Dilation and Asymmetric Relevance Gregory Wheeler A. Paul Pedersen - PowerPoint PPT Presentation

Dilation and Asymmetric Relevance Gregory Wheeler A. Paul Pedersen HMI Human & Machine Intelligence Frankfurt School of Finance & Management

This paper repairs characterization results in (Pedersen and Wheeler 2014; Pedersen and Wheeler 2015). Gregory Wheeler ¨ 3

P A openinterval P A P A P A . foreach B In other words, dilates A just in case the closed interval B is contained within the B Dilation Let B be a positive measurable partition of Ω . Say that B dilates A if each B P B : P p A | B q ă P p A q ď P p A q ă P p A | B q . Gregory Wheeler ¨ 4

openinterval interval is contained within the Dilation Let B be a positive measurable partition of Ω . Say that B dilates A if each B P B : P p A | B q ă P p A q ď P p A q ă P p A | B q . In other words, B dilates A just in case the closed “ ‰ P p A q , P p A q ` ˘ P p A | B q , P p A | B q foreach B P B . Gregory Wheeler ¨ 4

1 stochastic independence between events. otherwise. Dependence Given a probability function p on algebra A and events A , B P A , define: $ p p A X B q if p p A q p p B q ą 0 ; ’ & p p A q p p B q S p p A , B q : “ ’ % Thus the quantity S p is an index of deviation from Gregory Wheeler ¨ 5

Call the sets A lower probability space respectively, with radius . and upper neighborhoods of A conditional on B , lower B Given A B and Neighborhoods p Ω , A , P , P q , events A , B P A with P p B q ą 0, and ϵ ą 0, define: P p A | B , ϵ q : “ t p P P : | p p A | B q ´ P p A | B q| ă ϵ u ; P p A | B , ϵ q : “ t p P P : | p p A | B q ´ P p A | B q| ă ϵ u . Gregory Wheeler ¨ 6

Given lower probability space and upper neighborhoods of A conditional on B , Neighborhoods p Ω , A , P , P q , events A , B P A with P p B q ą 0, and ϵ ą 0, define: P p A | B , ϵ q : “ t p P P : | p p A | B q ´ P p A | B q| ă ϵ u ; P p A | B , ϵ q : “ t p P P : | p p A | B q ´ P p A | B q| ă ϵ u . Call the sets P p A | B , ϵ q and P p A | B , ϵ q lower respectively, with radius ϵ . Gregory Wheeler ¨ 6

Example by Michael Nielsen and Rush Stewart Corollary5.2of(PedersenandWheeler2014) that and Characterization B dilates A just in case there is p ϵ B q B P B P R B ` such P p A | B , ϵ B q Ď S ´ p A , B q P p A | B , ϵ B q Ď S ` p A , B q . Gregory Wheeler ¨ 7

and that Corollary5.2of(PedersenandWheeler2014) Stewart Example by Michael Nielsen and Rush Characterization 1 B dilates A just in case there is p ϵ B q B P B P R B ` such 0.8 P p A | B , ϵ B q Ď S ´ p A , B q 0.6 0.4 P p A | B , ϵ B q Ď S ` p A , B q . 0.2 0 0 0 0 0 0 1 2 . . 4 . 6 8 . Gregory Wheeler ¨ 7

S p A B S p A B S p A B S p A B S p A B 1 p S 1 and S A B p S p 1 A B I p p A B S 1 p A B S 1 A B Relevance S p and S p S p p A , B q : “ p p A X B q S p p A , B q : “ p p A X B q P p A q p p B q p p A q p p B q S p p A , B q : “ p p A X B q P p A q p p B q Gregory Wheeler ¨ 8

S and S p Relevance S p and S p S p p A , B q : “ p p A X B q S p p A , B q : “ p p A X B q P p A q p p B q p p A q p p B q S p p A , B q : “ p p A X B q P p A q p p B q S ` P p A , B q : “ t p P P : S p p A , B q ą 1 u ; ` P p A , B q : “ t p P P : S p p A , B q ą 1 u ; S ´ P p A , B q : “ t p P P : S p p A , B q ă 1 u ; S ´ P p A , B q : “ t p P P : S p p A , B q ă 1 u . I P p A , B q : “ t p P P : S p p A , B q “ 1 u . Gregory Wheeler ¨ 8

Theorem Let A be an event and B “ p B i q i P I be a positive measurable partition for a given set of probability functions P over an algebra. The following statements are equivalent (i) B dilates A; (ii) There exists ϵ ą 0 such that for every i P I: ` P p A | B i , ϵ q Ď S ´ P p A , B i q and P p H | B i , ϵ q Ď S P p A , B i q Gregory Wheeler ¨ 9

References Pedersen, A. P. and G. Wheeler (2014). Demystifying dilation. Erkenntnis 79 (6), 1305–1342. Pedersen, A. P. and G. Wheeler (2015). Dilation, disintegrations, and delayed decisions. In Proceedings of the 9th Symposium on Imprecise Probabilities and Their Applications (ISIPTA) , Pescara, Italy, pp. 227–236. Gregory Wheeler ¨ 10